Electron Spin Resonance at the Level of $10^4$ Spins Using Low Impedance Superconducting Resonators

We report on electron spin resonance measurements of phosphorus donors localized in a $200\mu {\mathrm{m}}^2$ area below the inductive wire of a lumped element superconducting resonator. By combining quantum limited parametric amplification with a low impedance microwave resonator design, we are able to detect around $2\times 10^4$ spins with a signal-to-noise ratio of 1 in a single shot. The 150 Hz coupling strength between the resonator field and individual spins is significantly larger than the 1–10 Hz coupling rates obtained with typical coplanar waveguide resonator designs. Because of the larger coupling rate, we find that spin relaxation is dominated by radiative decay into the resonator and dependent upon the spin-resonator detuning, as predicted by Purcell.

Figure 1

(a) Schematic of the experimental setup. The output field of the resonator is amplified by a JPA at base temperature, further amplified by a high-electron mobility transistor (HEMT) amplifier at 3.2 K and demodulated with an IQ mixer at room temperature to record and process the quadratures $I(t)$ and $Q(t)$ of the signal with FPGA electronics. (b) Optical micrograph of a resonator with low characteristic impedance $Z=\sqrt{L/C}\approx 8\mathrm{\Omega }$ that is side coupled to a coplanar transmission line for control and read-out. Coupling between the spins and the resonator field occurs below the inductive wire shown with higher magnification in (c). (d) Measurement (blue dots) and Lorentizan fit (red line) of the resonator transmission. (e) Measurement (blue dots) and Lorentzian fit (red line) of the JPA gain $G_{\omega }$. Measured noise power spectral density $S_{\omega }$ (black dots) in units of photons per Hz per second. (f) Simulated magnetic field distribution per current $b_1=B_1/I$ in the phosphorus doped region underneath the wire. (g) Simulated coupling strengths $g_0$ at three different depths below the wire, as indicated by the dashed lines in panel (f).

Figure 2

(a) Measured quadrature signal (dots) together with a numerical simulation of $⟨S^{-}(t)⟩$ (gray line). The echo duration is about $2T_2^{*}\approx 4\mu \mathrm{s}$. The data have been taken with the JPA turned off and by averaging over a thousand single shot time traces. (b) Integrated echo signal $I_{\text{echo}}$ as a function of magnetic field $B_0$. The asymmetry in the echo response for the two hyperfine peaks is predominantly explained by a change in the resonator frequency when sweeping $B_0$. (c) High resolution measurement of $I_{\text{echo}}$ around the low field electron spin transition (blue dots) and a Lorentzian fit (red line). (d) $I_{\text{echo}}$ as a function of $\pi$-pulse length ${\tau }_{\pi }$. The optimal ${\tau }_{\pi }=2.4\mu \mathrm{s}$. (e) $A_{\text{echo}}$ as a function of $2\tau$. Data are fit by an exponential decay (red line), with a best fit of $T_2=1.1\mathrm{ms}$.

Figure 3

Characteristic single shot time traces of the echo signal (black lines) and an average over 1000 single shot traces (red line) with the JPA turned off (a) and on (b). (c),(d) Individual measurement results for the filtered echo signal in the $IQ$ plane (blue dots) in comparison with the detection noise (red dots). From the distribution along the $I$ axis we extract a SNR of $\overline{I}/\mathrm{\Delta }I=10$ with the JPA turned on.

Figure 4

$T_1$ as a function of $\delta =E_z/\hslash -{\omega }_{\text{res}}$. (a) Two characteristic data sets from which the $T_1$ times are extracted. Inset: Population recovery pulse sequence. (b) Comparison between the measured lifetimes (blue dots) and the Purcell prediction (red line) assuming $g_0/2\pi =150\mathrm{Hz}$ and the independently measured $\kappa /2\pi =1.6\mathrm{MHz}$. Each data point results from the average of two values independently fitted to data sets as shown in panel (a). Error bars indicate the standard error extracted from the exponential fit.